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Express the number as a ratio of intergers.

$ 10.1 \overline {35} = 10.135353535 . . . $

$10.1 \overline{35}=\frac{5017}{495}$

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Oregon State University

Harvey Mudd College

University of Nottingham

Idaho State University

Let's express this. A number that's given to us is the ratio of imagers. So that's a fraction of the form p over Q or peeing cure imagers. This means that their numbers of the form like zero one and two and so on and also the negative is a swell. No fractions or pies or anything like that. So here, let's start off by just pulling out the ten in the point one. Because the pattern really starts when we get to the three five. That's the route of the repeated part. Notice that the one is not included and you only see one just one time. Then, after that, let's switch the color here. Then I have removed this first spot because I already have the one there. And then after that, I'LL write three five and then so that deals with these two right here and then after that, the first three spots have already been used, so I'll have to add three zeros in here for the first three spots. That's one one took care of the first pot than the three for the second of five for the third, and then now we have another three five here so that a little here and we could even do one more. So now this time we've used up five of the spot's already. This was the first over here, the one than to three and then four in five. So we have to have zeros there. There's already being used, and then we have another three five and and so on, and we could see each time that we're getting to new zeros. And that's what's important here. Two more zeros. This means that we're multiplying by one or ten squared each time we go to the right. So this is geometric and we're multiplying by one over one hundred. But the first term is in the thousand spot, so we will have to be careful here. So this is point one. So I'm just rewriting these, and then now let's start writing these fractions. So the first one that's thirty five, but that's over a thousand told me. Write ten cute, then thirty five and Sensei multiplying by one over ten square. There's I get a five three plus two, and then we just keep adding to to that exploded in the denominator, and that's our are geometric series on the inside, so we know for a geometric. Siri's just this part right here. This is the geometric part, and we know that the sum we know first of all, that this converges because this our value satisfies the inequality. Absolute value is less than one silk emerges. Then the sum is you take the first term of the series and then divided by one minus. R. Our first term is thirty five over ten. Cubed divide that by one minus R. So that's what we have there. And then let's go ahead and simplify this. So thirty five over ten. Cute. And that's ninety nine over ten square on the bottom that ends up being thirty five over nine nine nine, nine zero. So that will. This fraction here will replace this big geometric some here. So we have ten still us three, right? This is a fraction, too. Let's write that has won over ten and then this geometric some we just found the fraction for after we use the formula for a geometric series. Now we have three fractions to add, and we add all these together five o one seven in the numerator, all over four hundred and ninety five in the denominator. And that's our final answer